Math, asked by loga0947, 2 months ago

integral x+1×dx/x^2-1 using integration by parts​

Answers

Answered by milk5000milk
0

Answer:

Answer:

Given :-

The nth term of an AP is 3, 8, 13, 18 is 73.

To Find :-

What is the value of n or number of terms of an AP.

Formula Used :-

\clubsuit♣ General term or nth term of an AP Formula :

\begin{gathered}\bigstar \: \: \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\: \: \: \bigstar\\\end{gathered}

a

n

=a+(n−1)d

where,

\sf a_na

n

= nth term of an AP

a = First term of an AP

n = Number of terms of an AP

d = Common Difference of an AP

Solution :-

First, we have to find the common difference (d) :-

Given :

a₁ = 3

a₂ = 8

According to the question :

\begin{gathered}\implies \bf Common\: Difference(d) =\: a_2 - a_1\\\end{gathered}

⟹CommonDifference(d)=a

2

−a

1

\implies \sf Common\: Difference(d) =\: 8 - 3⟹CommonDifference(d)=8−3

\implies \sf\bold{\purple{Common\: Difference(d) =\: 5}}⟹CommonDifference(d)=5

Hence, the common difference or d is 5 .

Now, we have to find the value of n :

Given :

nth term \sf (a_n)(a

n

) = 73

First term (a) = 3

Common Difference (d) = 5

According to the question by using the formula we get,

\dashrightarrow \bf a_n =\: a + (n - 1)d⇢a

n

=a+(n−1)d

\dashrightarrow \sf 73 =\: 3 + (n - 1)5⇢73=3+(n−1)5

\dashrightarrow \sf 73 - 3 =\: (n - 1)5⇢73−3=(n−1)5

\dashrightarrow \sf 70 =\: (n - 1)5⇢70=(n−1)5

\dashrightarrow \sf \dfrac{\cancel{70}}{\cancel{5}} =\: (n - 1)⇢

5

70

=(n−1)

\dashrightarrow \sf \dfrac{14}{1} =\: (n - 1)⇢

1

14

=(n−1)

\dashrightarrow \sf 14 =\: n - 1⇢14=n−1

\dashrightarrow \sf - n =\: - 1 - 14⇢−n=−1−14

\dashrightarrow \sf {\cancel{-}} n =\: {\cancel{-}} 15⇢

n=

15

\dashrightarrow \sf\bold{\red{n =\: 15}}⇢n=15

\therefore∴ The value of n or number of terms of an AP is 15 .

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EXTRA IMPORTANT FORMULA :-

\diamond⋄ General term or nth term of an AP Formula :

\begin{gathered}\bigstar \: \: \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\: \: \: \bigstar\\\end{gathered}

a

n

=a+(n−1)d

where,

\sf a_na

n

= nth term of an AP

a = First term of an AP

n = Number of terms of an AP

d = Common Difference of an AP

\diamond⋄ Sum of nth term of an AP Formula :

\begin{gathered}\bigstar \: \: \sf\boxed{\bold{\pink{S_n =\: \dfrac{n}{2}\bigg\lgroup 2a + (n - 1)d\bigg\rgroup }}}\: \: \: \bigstar\\\end{gathered}

S

n

=

2

n

2a+(n−1)d

where,

\sf S_nS

n

= Sum of nth term of an AP

n = Number of terms of an AP

a = First term of an AP

d = Common Difference of an AP

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